An axiom is a sentence or proposition that is accepted as the first and last line of a one-line proof and is considered as obvious or as an initial necessary consensus for the theory building or acceptation. Therefore, it is taken for granted as true, and serves as a starting point for deducing and inferencing other truths.
This is a wrong definition. I was writing a correct definition to Wikipedia in Finnish but theologians and philosophers were changing it. The reasons were ideological.
The axiom is never taken for granted as true because it is a part of the definition of the concepts.
An axiom is a part of the definition of the concepts in the sentence.
We can learn the language using many methods of definition:
We learn words using all methods of definition.
In certain epistemological theories, an axiom is a self-evident truth upon which other knowledge must rest, and from which other knowledge is built up. An axiom in this sense can be known before one knows any of these other propostions. Not all epistemologists agree that any axioms, understood in that sense, exist.
This is wrong because
1. It is wrong to use the word “axiom” for “self-evident truths”.
2. There are no self-evident truths.
Of course it is possible use axiomatic way to define the concepts of the epistemology. For example Alvin I. Goldman uses a weak definition of knowledge (in Knowledge in a Social World):
“...knowledge is here understood in the ‘weak’ sense of true belief”.
This is partial definition of the words knowledge, true anf belief.
This definition assumes that you know the meanings of the words is, here, understand, weak and sense.
In logic and mathematics, an axiom is not necessarily a self-evident truth, but rather a formal logical expression used in a deduction to yield further results. To axiomatize a system of knowledge is to show that all of its claims can be derived from a small set of sentences that are independent of one another. This does not imply that they could have been known independently; and there are typically multiple ways to axiomatize a given system of knowledge (such as arithmetic). Mathematics distinguishes two types of axioms: logical axioms and non-logical axioms.
This is not exact. Axioms are implicit definitions of the fundandamental concepts.
In Euclidean geometry following sentences will define the concepts point, line and plane:
“For every two points A, B there exists a line a that contains each of the points A, B.
For every two points A, B there exists no more than one line that contains each of the points A, B.
There exists at least two points on a line. There exist at least three points that do not lie on a line.
For any three points A, B, C that do not lie on the same line there exists a plane α that contains each of the points A, B, C. For every plane there exists a point which it contains.
For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C.
If two points A, B of a line a lie in a plane α then every point of a lies in the plane α.
If two planes α, β have a point A in common then they have at least one more point B in common.
There exist at least four points which do not lie in a plane”
The main reason to use false definitions of the concepts is theological or philosophical. Theologians can use for example following axioms:
There is only one god.
The god is all-mighty.
The god is all knowing.
The god is all-good.
This is one of the definitions of the Christian god. But there is no such god. We can use axioms to define being, but it is possible, that there is no such being.
There are mathematicians who think that the Euclidean definition of the line is wrong. It is not wrong because it is the definition of the Euclidean line. Non-Euclidean geometries have different axioms.
The publishing of the dictionary is one part of the use of the power.
Our use of the language is often weak. If somebody asks what we mean when we say for example “science” it is probable that we can not give a complete answer.
As atheists we can not accept the ideological use of the weak language. Most of the theology and the philosophy is opinions using emotional and weak language.